The research results are as follows :(1) For Markov chains with discrete-time and continuous-time, which correspond to a generalization of the Ehrenfest urn model, we have determined the structure of transition probabilities and stationary distributions. The results have developed into applications to reliability theory and queueing system in operations research.(2) For stochastic processes with fast-motion and delayed motion, which are tagged to multi-particle and considered as solutions of linear stochastic differential equations with mean-field interaction of the McKean type under singular perturbation, we have derived and identified the limit process and clarified the effect of interacting force.(3) For Markov processed, which are solutions of the nonlinear 3-dimensional stochastic differential equations of the Lorenz type, we have obtained a stochastic Duffing oscillator with random force as a limit process by a suitable space-time transformation and an asymptotic analysis.(4) We have investigated the above-cited results by computer network and information processing, that have developed into applications to noise analyses, such as image compression, population dynamics, economic time-series, information security and random fractal.(5) We have presented the above-cited results at the Japan SIAM meeting, the RIMS(Kyoto Univ.) workshop and the Japan IMA meeting.(6) We are going to extend the above-cited results to the theory and application of discrete approximation for stochastic differential equations with self-similar noises and data compression of fractal images in multi-media.